procedure? the HCF of the following by factorization. \( \begin{array}{ll}(x+y)^{2}, x^{2}-y^{2} & \text { 3. }(a-b)^{3}, a^{2}-2 a b+b^{2} \\ a^{3} b-a b^{3}, a^{5} b^{2}-a^{2} b^{5} & \text { 5. } x^{2}-49, x^{2}-4 x-21 \\ 12 x^{2}+x-1,15 x^{2}+8 x+1 & \text { 7. } c^{2} x^{2}-d^{2}, a c x^{2}-b c x \\ m^{2}-n^{2}, m^{4}-n^{4}, m^{6}-n^{6} & \\ a x^{2}+2 a^{2} x+a^{3}, 2 a x^{2}-4 a^{2} x-6 a^{3}, 3\left(a x+a^{2}\right)^{2}\end{array} \)

To find the highest common factor (HCF) of the given pairs of expressions by factorization, we will factor each expression and then identify the common factors. Let's go through each pair step by step. ### 1. \( (x+y)^{2}, x^{2}-y^{2} \) - **Factor \( (x+y)^{2} \)**: \[ (x+y)^{2} = (x+y)(x+y) \] - **Factor \( x^{2}-y^{2} \)** (difference of squares): \[ x^{2}-y^{2} = (x-y)(x+y) \] - **Common factor**: The common factor is \( (x+y) \). ### 2. \( (a-b)^{3}, a^{2}-2ab+b^{2} \) - **Factor \( (a-b)^{3} \)**: \[ (a-b)^{3} = (a-b)(a-b)(a-b) \] - **Factor \( a^{2}-2ab+b^{2} \)** (perfect square): \[ a^{2}-2ab+b^{2} = (a-b)^{2} \] - **Common factor**: The common factor is \( (a-b)^{2} \). ### 3. \( a^{3}b-ab^{3}, a^{5}b^{2}-a^{2}b^{5} \) - **Factor \( a^{3}b-ab^{3} \)**: \[ ab(a^{2}-b^{2}) = ab(a-b)(a+b) \] - **Factor \( a^{5}b^{2}-a^{2}b^{5} \)**: \[ a^{2}b^{2}(a^{3}-b^{3}) = a^{2}b^{2}(a-b)(a^{2}+ab+b^{2}) \] - **Common factor**: The common factor is \( ab(a-b) \). ### 4. \( x^{2}-49, x^{2}-4x-21 \) - **Factor \( x^{2}-49 \)** (difference of squares): \[ x^{2}-49 = (x-7)(x+7) \] - **Factor \( x^{2}-4x-21 \)**: \[ x^{2}-4x-21 = (x-7)(x+3) \] - **Common factor**: The common factor is \( (x-7) \). ### 5. \( 12x^{2}+x-1, 15x^{2}+8x+1 \) - **Factor \( 12x^{2}+x-1 \)**: Using the quadratic formula or factoring: \[ 12x^{2}+x-1 = (3x-1)(4x+1) \] - **Factor \( 15x^{2}+8x+1 \)**: Using the quadratic formula or factoring: \[ 15x^{2}+8x+1 = (3x+1)(5x+1) \] - **Common factor**: There are no common factors. ### 6. \( c^{2}x^{2}-d^{2}, acx^{2}-bcx \) - **Factor \( c^{2}x^{2}-d^{2} \)** (difference of squares): \[ c^{2}x^{2}-d^{2} = (cx-d)(cx+d) \] - **Factor \( acx^{2}-bcx \)**: \[ acx^{2}-bcx = x(acx-bc) = x(c(ax-b)) \] - **Common factor**: The common factor is \( x \). ### 7. \( m^{2}-n^{2}, m^{4}-n^{4}, m^{6}-n^{6} \) - **Factor \( m^{2}-n^{2} \)** (difference of squares): \[ m^{2}-n^{2} = (m-n)(m+n) \] - **Factor \( m^{4}-n^{4} \)** (difference of squares): \[ m^{4}-n^{4} = (m^{2}-n^{2})(m^{2}+n^{2}) = (m-n)(m+n)(m^{2}+n^{2}) \] - **Factor \( m^{6}-n^{6} \)** (difference of squares